Do you know the vertical line test for a function?
What would more than two horizontal asymptotes imply?
Have a look at this.
Your question has nothing to do with "multivariable or vector Calculus", it has only to do with the definition of "horizontal asymptote".
If you want to ask your question in terms of multiple independent variables, you will have to define "horizontal" an "horizontal asymptote" in multiple variables.